Sarah joins a gym. After 1 month she has paid £45 in total, and this total rises by £25 each further month. Write a formula for the total Sarah has paid aft...

This question tests building an \(n\)th term formula for a real cost sequence, using it to compare against another sequence, and reasoning about growth rates without further calculation.

(a) After 1 month Sarah has paid \(\pounds45\), and the total rises by \(\pounds25\) each further month, so the total after \(n\) months is:

[3]

(b) Substituting \(n=12\):

Sarah has paid \(\pounds320\) after 12 months. [1]

(c) Setting Sarah's total equal to Tomi's:

[3]

At \(n=3\), both totals are \(25(3)+20 = 95\) and \(30(3)+5=95\), so their totals are equal after \(3\) months, both \(\pounds95\).

(d) After 10 months: Sarah has paid \(25(10)+20=270\), while Tomi has paid \(30(10)+5=305\). Since \(270 < 305\), Sarah's gym is cheaper after 10 months. [2]

(e) Sarah's total increases by \(\pounds25\) every month (her common difference), while Tomi's increases by \(\pounds30\) every month. Since Tomi's monthly increase is larger, Tomi's total grows faster with every extra month, which is why Tomi's total is lower before month 3 but overtakes and stays higher afterwards, without needing any further arithmetic to justify it. [2]

The common difference of a linear cost model is its growth rate; comparing common differences directly (rather than recalculating totals) is the efficient way to reason about which of two competing linear models will eventually be more expensive.